The foot of a ladder leaning against a wall of length 5 metres rests on a level ground ` 5 sqrt(3)` metres from the base of the wall. The angle of inclination of the ladder with the ground is

Options

A. `60^(@)`
B. ` 50^(@)`
C. `40^(@)`
D. `30^(@)`

Answer

To find the angle of inclination of the ladder with the ground, we can follow these steps:

### Step 1: Understand the Problem
We have a ladder leaning against a wall. The length of the ladder (hypotenuse) is 5 meters, and the distance from the base of the wall to the foot of the ladder (base) is \(5\sqrt{3}\) meters. We need to find the angle of inclination (θ) of the ladder with respect to the ground.

### Step 2: Draw the Diagram
Draw a right triangle where:
- The wall is represented as the vertical side (perpendicular).
- The ground is represented as the horizontal side (base).
- The ladder is represented as the hypotenuse.

Label the points:
- A (top of the wall)
- B (foot of the ladder)
- C (base of the wall)

### Step 3: Identify the Sides of the Triangle
- Length of the ladder (AB) = 5 meters (hypotenuse)
- Distance from the base of the wall to the foot of the ladder (BC) = \(5\sqrt{3}\) meters (base)
- Height of the wall (AC) = unknown (perpendicular)

### Step 4: Use the Pythagorean Theorem
Since we have a right triangle, we can use the Pythagorean theorem:
\[ AB^2 = AC^2 + BC^2 \]
Substituting the known values:
\[ 5^2 = AC^2 + (5\sqrt{3})^2 \]
\[ 25 = AC^2 + 75 \]
\[ AC^2 = 25 - 75 \]
\[ AC^2 = -50 \]
This indicates that we made a mistake in our calculations. Let's find the height (AC) using the tangent function instead.

### Step 5: Use the Tangent Function
In a right triangle, the tangent of an angle (θ) is given by:
\[ \tan(θ) = \frac{\text{perpendicular}}{\text{base}} = \frac{AC}{BC} \]
Here, we know:
- Perpendicular (AC) = 5 meters (height of the wall)
- Base (BC) = \(5\sqrt{3}\) meters

So,
\[ \tan(θ) = \frac{5}{5\sqrt{3}} = \frac{1}{\sqrt{3}} \]

### Step 6: Find the Angle
Now, we need to find the angle θ such that:
\[ \tan(θ) = \frac{1}{\sqrt{3}} \]
From trigonometric values, we know:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
Thus,
\[ θ = 30^\circ \]

### Conclusion
The angle of inclination of the ladder with the ground is \(30^\circ\).

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